It is essential to maintain a "phone book" of domain names with their matching numerical (IP) addresses for the world-wide Domain Name System (DNS) to work. A world-wide decentralized hierarchical naming network processes all changes, propagating each new domain or change to all it's members. This avoids having only one, vulnerable, centralized database.
Independent authorized vendors, called Registrars, process domain name searches, requests, transfers and other related services, for a fee, usually about $15/yr, and update the DNS network, which can take up to 48 hours to propagate to your computer's DNS service provider.
In general the site owner is responsible for purchasing and annualy renewing the domain name with a Registrar. In some cases, like an all-in-one WordPress package, this is part of the package, which makes signing up simple, and canceling a hosting service without losing your domain name complicated and risky. If you may ever want to migrate your domain to a new hosting provider, or transfer your domain name to a new registrar, opt for an independent registrar, which garauntees your ownership of your domain name even if you transfer to a different registrar or hosting provider.
If you want me to manage your registrar account after you have signed up with your credit card and email address you would need to trust me with your registrar logins (user name and password), which you should record in your own records as well, but not your credit card info, which I don't want. That would give me access to set up automatic annual renewals, do transfers to other registrars or hosting providers, and other registrar services.
I have used the Registrar DirectNIC for years and recommend them.